These constraints may not lead to physically acceptable solutions. However, in the three qubit case, one finds that states with maximal entanglement measure does exist.
We start with a canonical form of the three qubits and apply these constraint equations. Acín et al. [1]} have shown that any pure three qubit can be reduced to a canonical form, having five amplitudes and one relative phase using the generalised Schmidt decomposition [2]}. Any generic three qubit state can be reduced to several classes of canonical forms using five local base product states (LBPS), while preserving nonlocal properties of the state. Owing to different degrees of orthogonality, there exists three inequivalent classes of five LBPS.
We begin with the general state built using the canonical forms from these LBPS classes. Once a maximally entangled state is obtained, we perform local unitary (LU) transformations to find other states that form a complete basis in the Hilbert space of three qubit system, as LU operations do not affect entanglement of the state [3]}.
